Path-connectedness for finite topological spaces

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1. May 16, 2017

Wendel

1. The problem statement, all variables and given/known data
I'm trying to understand the intuition behind path-connectedness and simple-connectedness in finite topological spaces. Is there a general methodology or algorithm for finding out whether a given finite topological space is path-connected?

2. Relevant equations
how can I determine which of the following topologies are path-connected?
for the three-element set:
1. {∅,{a,b,c}}
2. {∅,{c},{a,b,c}}
3. {∅,{a,b},{a,b,c}}
4. {∅,{c},{a,b},{a,b,c}}
5. {∅,{c},{b,c},{a,b,c}}
6. {∅,{c},{a,c},{b,c},{a,b,c}}
7. {∅,{a},{b},{a,b},{a,b,c}}
8. {∅,{b},{c},{a,b},{b,c},{a,b,c}}
I won't list them out, but for the four-point set there are 33 inequivalent topologies. One of which is the pseudocircle X={a,b,c,d} which has the topology {{a,b,c,d},{a,b,c},{a,b,d},{a,b},{a},{b},∅}.

3. The attempt at a solution
The discrete topology is totally disconnected. Any map from the unit interval to the indiscrete topology is continuous, so it must be path-connected. Furthermore the particular point topology is path-connected. The pseudocircle is clearly path-connected since the continuous image of a path-connected space is path-connected. Furthermore it is not simply connected. From Wikipedia, connectedness and path-connectedness are the same for finite topological spaces.
Thank you, apologies for the long post.

2. May 16, 2017

andrewkirk

Since connectedness is easier to test than path-connectedness, for a small finite topology, why not just test each of the topologies for connectedness? All you need to do is, for each topology, look at all pairs of open sets, excluding the empty and universal set. If one or more of those pairs is a partition of the point set, the topology is disconnected, otherwise it is connected. If you have $n$ non-empty, non-universal open sets in the topology, there are only ${}^nC_2$ pairs of sets to examine, which is only 6 for the richest of the above topologies (the last one).

3. May 18, 2017

Wendel

Thank you, it makes more sense now. However, I can't help but wonder, given a pair of points x,y∈X where X is a finite topological space, how to determine what other points in X must lie "between" x and y in order for it to be a path from x to y. Generally, the path wouldn't be surjective. I'm trying to reconcile my intuitive grasp of paths and homotopy with the more abstract notion of a topology.

Last edited: May 18, 2017
4. May 18, 2017

andrewkirk

Well a 'path' will be a function from the interval $I=[0,1]$ to the set $S$ on which the topology is imposed.
Just as in the continuous case, there will usually be multiple different possible paths between two points.
For a function $f$ to qualify as a 'path' between $x$ and $y$, both in $S$, all we require is that
1. $f(0)=x$
2. $f(1)=y$ and
3. $f$ is continuous.
The last one means that that for every open subset of $S$, the inverse image $f^{-1}(S)$ must be open in $I$.

For the trivial topology, any $f:I\to S$ will be continuous, so if it satisfies 1 and 2 it will be a path. So any point in $S$ can be on a path between $x$ and $y$.

On the other hand if $S$ has the discrete topology, so that every subset is open, no function $f:I\to S$ will be continuous, so there are no paths, unless $x=y$, in which case there is one special function $f$ that is continuous and is a path. Can you figure out why this is the case?

Paths in finite topologies will typically be step functions - or at least the ones we can easily visualise will be. So they'll sit on one point in $S$ until the input number gets to a certain value, at which they'll suddenly jump to another point. See if you can use that to work out a path between $a$ and $c$ in topology 2 above. Interestingly, we can only get paths going in one direction. Which direction, a to c, or c to a?